To build the reduced spaces \(V^{\text{f},rb}_{k,T}\) , we use a standard weak greedy [1]} approach in order to
minimize the model order reduction error \((v_H) - (v_H)\)  for all \(\mu \in \mathcal {P}\)  and \(v_H \in V_H\) .
To this end, we choose a target error tolerance \(\varepsilon _1\)  and an appropriate training set of parameters
\(\mathcal {P}_\textnormal {train}\) , over which we estimate the maximum reduction error.
The initial reduced space \(V^{\text{f},rb}_{k,T}\)  is chosen as the zero-dimensional space.
Then, in each iteration, the reduction error is estimated with \(\eta _{T,\mu }((v_H))\)  for all \(\mu \in \mathcal {P}_\textnormal {train}\)  and \(\phi _{i_{T,j}}\) , and a pair \(\mu ^*\) , \(\phi _{i_{T,j^*}}\)  maximising the estimate is selected.
Thanks to the offline-online decomposition of \(\eta _{T,\mu }\)  this step does not involve any high-dimensional
computations, so \(\mathcal {P}_\textnormal {train}\)  can be chosen large.
Since (REF ) is linear, it suffices to consider the basis functions \(\phi _{i_{T,j}}\)  as \(v_H\) .
After \(\mu ^*\) , \(j^*\)  have been found, \([T][\mu ^*](\phi _{i_{T,j^*}})\)  is computed.
\(V^{\text{f},rb}_{k,T}\)  is extended with this solution snapshot, and the offline-online decompostion for this extended reduced
space is computed.
The iteration ends, when the maximum estimated error drops below \(\varepsilon _1\) .
A formal definition of the procedure is given in Algorithm REF .
